FIG 17 - uploaded by Evgeny A. Demekhin
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(Color online) Final stage of evolution. For the parameters and notation, see Fig. 12 (e) and the point E of the map 8.
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The hydrodynamics and electrostatics of imperfect electric membranes are examined numerically. The investigation is based on the Nernst–Planck–Poisson–Stokes system of equations. A three-layer geometry, electrolyte–nanoporous membrane–electrolyte, is considered. The threshold of the electrokinetic instability of the one-dimensional quiescent state...
Citations
... Течение электролита в мембранах обычно описывается уравнениями Пуассона и Дарси [9][10][11]. В достаточно широких каналах используется подход, основанный на предположении об электронейтральности электролита вне тонких (до 100 нм) дебаевских слоёв [12]. ...
The behavior of a diluted electrolyte in a system of joint microchannel and nanochannel with charged dielectric walls under the action of external potential difference and external pressure is investigated numerically. The surface charge on the nanochannel walls prevents the ions of corresponding polarity from passing through it. Consequently, the system in question acquires ion-selective properties and can, under certain assumptions, be viewed as a fragment of an ion-selective membrane, including one synthesized by creating nanopores in a dielectric material. Such systems are used in experiments to control the movement of charged particles through concentration polarization. The objective of the work is to investigate the influence of a single pore on electrolyte flow and the possibilities to control that flow by changing the geometric and physical properties of the pore. The investigation relies on the specially developed simplified models based on cross-section-averaged Nernst-Planck, Poisson and Stokes equations that are subsequently reduced to a single nonlinear differential equation. The simplified models allow identifying the impact of different physical mechanisms of electrolyte movement: pressure-based (generated by the external mechanical action) and electroosmotic (generated by the electric field). A finite-difference method with semi-implicit time integration is used for the numerical solution of equations. It has been found that the behavior of the system qualitatively matches the behavior of a cell based on a non-ideally-selective ion-exchange membrane. In particular, the model correctly predicts the underlimiting and limiting electric current regimes, as well as vortex formation near the nanochannel inlet due to concurrency between electrolyte movement mechanisms. The proposed models can be extended to describe a channel with arbitrary geometry and an electrolyte with arbitrary number of charged species.
... When the voltage is increased beyond a critical value, current does not increase monotonously with the increasing trend of applied voltage, this indicates the system enters the limiting regime. In the overlimiting regime, the current increases sharply again with the applied voltage is further increased beyond another critical value [49,50]. The system cannot theoretically sustain a current greater than a limiting value corresponds to the complete depletion of ion concentration at the electrolyte-membrane interface, this is according to the assumption of local electroneutrality (LEN) [51]. ...
In electromembrane processes employing the ion concentration polarization phenomenon, such as electrodialysis (ED), ion concentration polarization (ICP) desalination, and reverse electrodialysis, the ion-selective membrane (ISM) has been modeled with an assumption of an ideal ISM permits only the passage of counter-ions with all the co-ions blocked. However, in the works that study the effects of co-ion flux on the response of system, this assumption has been questioned about the adequate answer to the simulation of the characteristic properties of the membrane. In this work, we evaluate the three models which are used widely in modeling ISM including surface-charged, space-charged and nanochannel array model. The results show that the nanochannel array and space-charged models, the alternative model for the model ISM in the case of the studies which consider the effects of co-ion leakage on the performance of the system, sufficiently mimic the key characteristic of the membrane. Their current-voltage response (I-V) curve conforms to a typical one, yet they differ due to the varying co-ion fluxes. The nanochannel array model, which is the direct model of the actual ISM, closely mimics the seed electroconvection vortices near the membrane, which does not appear in the surface-charged and space-charged model. This model is computationally expensive because of resolving the nonlinearity of the variables near the entrance of nanochannels, so it is not the appropriate model to employ in a study that considers the micrometer or millimeter order scale system. Finally, the space-charged model is more appropriate to mimic the ISM but requires more careful consideration of the simplification condition. By investigating the volume charge density of the membrane in the space-charged model, we suggest a value that brings the most similar I-V curve to that of the nanochannel array model.
... In the works of Rubinstein I. [36][37][38], Demekhin E.A., Kalaidin E.N. [39,40], the problems of occurrence and stability of electroconvection in micro-and nanofluidics in the absence of forced convection are investigated on the basis of mathematical modeling. ...
This article considers a theoretical analysis of the influence of the main coupled effects and spacers on the transfer of salt ions in electromembrane systems (EMS) using a 2D mathematical model of the transfer process in a desalting channel with spacers based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The basic patterns of salt ion transport have been established, taking into account diffusion, electromigration, forced convection, electroconvection, dissociation/recombination reactions of water molecules, as well as spacers located inside the desalting channel. It has been shown that spacers and taking into account the dissociation/recombination reaction of water molecules significantly change both the formation and development of electroconvection. This article confirms the fact of the exaltation of the limiting current studied by Harkatz, where it is shown that the current (flux) of salt ions increases when the dissociation reaction begins by a certain value called the exaltation current, which is proportional to the flow of water dissociation products. A significant combined effect of electroconvection and dissociation/recombination reactions as well as the spacer system in the desalting channel on the transport of salt ions are shown. The complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization and spacers in the desalting channel are also considered in the work.
... Subsequently, this microscale nonequilibrium vortex aroused the interest of many physicists, and various studies on the nonlinear effects of EC were carried out, including the spatiotemporal evolution of the ion depletion layer, flow patterns, and OLC. Demekhin et al. [14][15][16] carried out a series of numerical studies on nonequilibrium EC, and found that the vortex pairs become chaotic vortices as the voltage increases, thus exhibiting linear and nonlinear evolutionary laws. Mani and other collaborators [17][18][19] performed detailed statistical analysis on the fluctuation characteristics of the fluid and ion concentrations. ...
Electroconvection has the potential to be applied in electrochemical technologies such as electrodialysis and energy storage, and has thus aroused considerable research interest. This paper describes the direct numerical simulation (DNS) of the dimensionless Poisson–Nernst–Planck and Stokes equations for electroconvection to determine why the dimensionless thin Debye layer in existing simulations does not match the results of canonical experiments. Our DNS results show that the discrepancy between the simulation results and the experimental data is mainly caused by differences in the structural characteristics of the extended space charge layer. A dimensionless thin Debye layer matching those in canonical experiments enhances the driving force of the extended space charge layer, resulting in massive vortices near the permselective membranes that cause the electroconvective flow to transition from the steady state to time-dependent spatiotemporal dynamics. Our DNS results show that choosing the thickness of the dimensionless thin Debye layer to be consistent with canonical experiments is a key factor in the high-precision quantitative analysis of electroconvection characteristics such as the vortex height, dynamic evolution, and pattern formation. These results provide important guidance for the design and instability control of microfluidic chips.
... If the voltage is further increased beyond another critical value, a sharp increase in current occurs and the system enters the overlimiting current (OLC) regime. 27,28 A great number of theoretical studies have been conducted to reveal the ion transport mechanism behind the transport behaviors in these three regimes. The behaviors in Ohmic and LC regimes could be explained in terms of ion transport in stationary and neutral solutions. ...
Ion selective membrane (ISM) is widely used in electrochemical engineering and micro-fluidic processes, yet accurate modeling of the ISM
is still challenging due to many scientific issues. So far, assumptions on the “ideal ISM” have been used in most simulation studies involving
ion transport and electrokinetic flow in ion concentration polarization systems, but the validity or accuracy of those assumptions has never
been investigated. In this paper, using a two-dimensional nanochannel system with practical significance, we verify the validity of the ideal
ISM model by making comparisons between the idealized ISM system and a more realistic permselective nanochannel system in terms of
simplifications over the electrical potential, counter-ion concentration, and zero co-ion flux. Our results show that the simplifications of fixed
voltage and fixed counter-ion concentration in the ideal ISM model are largely accurate in most situations, especially under high applied
voltage and/or with high charge density inside the ISM. However, zero co-ion flux simplification is not exactly accurate in most occasions.
Significant errors may be incurred by the zero co-ion flux assumption when steady state solutions are sought using the ISM model. Some
discussions over the influences of structures of the nanochannel system are also added. The obtained results will help in obtaining detailed
understanding of the transport features inside the nanoporous ISM, especially when the comparison between simulation and experimental
data is necessary.
... One important active technique to enhance the heat transfer is to employ an electrohydrodynamically induced flow motion. The electrohydrodynamics (EHD) based heat transfer enhancement is characterized by quick response, low power consumption and offers design flexibility with no moving parts ( Atten, 1996;Suh, 2012;Pérez et al., 2014;Guan et al., 2018;Demekhin et al., 2018 ). EHD has been proven to be an effective heat transfer enhancement technique in both single and multi-phase systems ( Rashidi et al., 2017;Allen and Karayiannis, 1995 ). ...
Recent experimental results demonstrate that electric field can effectively decrease the melting time of dielectric Phase Change Materials (PCMs). In this study, a Finite-Volume Method (FVM) based numerical model for the solid-liquid phase change heat transfer of dielectric PCM under the influence of electric field is presented. Fully coupled governing equations of electric potential, charge transport, Navier-Stokes equations, and the energy equation are implemented in the finite-volume framework of OpenFOAM®. The numerical model is first validated against the analytical solutions for several test cases in the hydrostatic regime. Results from the numerical model exhibit good agreement with the analytical solutions. The numerical model presented in this work is capable of capturing the sudden step change in the charge density distribution and electric field due to the discontinuity of the physical properties at the interface. A numerical analysis of EHD assisted melting of a dielectric PCM inside a rectangular cavity is considered. Effects of electric Rayleigh number T and Stefan number St on the rate of melting are discussed. The transient evolution of the EHD assisted melting process which includes different flow stages is analyzed. It is found that the electric Rayleigh number T has a notable effect on the rate of melting and its influence is more pronounced at lower values of St. A maximum of 56.10% reduction in total melting time is achieved at T=3000 and St=0.01, for the flow configuration considered here.
... The properties of an imperfect membrane can be very different from its perfect analog. A mathematical formulation for the imperfect ion-selective membrane was first put forth in [21] and exploited in the works in [22][23][24][25][26]. In all the mentioned works only the electrostatic properties of an electric membrane were taken into account and the fluid inside the porous membrane is considered immobile. ...
Numerical investigation of the underlimiting, limiting, and overlimiting current modes and their transitions in imperfect ion-selective membranes with fluid flow through permitted through the membrane is presented. The system is treated as a three layer composite system of electrolyte-porous membrane-electrolyte where the Nernst-Planck-Poisson-Stokes system of equations is used in the electrolyte, and the Darcy-Brinkman approach is employed in the nanoporous membrane. In order to resolve thin Debye and Darcy layers, quasi-spectral methods are applied using Chebyshev polynomials for their accumulation of zeros and, hence, best resolution in the layers. The boundary between underlimiting and overlimiting current regimes is subject of linear stability analysis, where the transition to overlimiting current is assumed due to the electrokinetic instability of the one-dimensional quiescent state. However, the well-developed overlimiting current is inherently a problem of nonlinear stability and is subject of the direct numerical simulation of the full system of equations. Both high and low fixed charge density membranes (low- and high concentration electrolyte solutions), acting respectively as (nearly) perfect or imperfect membranes, are considered. The perfect membrane is adequately described by a one-layer model while the imperfect membrane has a more sophisticated response. In particular, the direct transition from underlimiting to overlimiting currents, bypassing the limiting currents, is found to be possible for imperfect membranes (high-concentration electrolyte). The transition to the overlimiting currents for the low-concentration electrolyte solutions is monotonic, while for the high-concentration solutions it is oscillatory. Despite the fact that velocities in the porous membrane are much smaller than in the electrolyte region, it is further demonstrated that they can dramatically influence the nature and transition to the overlimiting regimes. A map of the bifurcations, transitions, and regimes is constructed in coordinates of the fixed membrane charge and the Darcy number.
... It is noted that the plateau in Di reminds the over-limiting current for ion transport through nano/microchannels. [65][66][67] However, the underlying mechanisms are different. The former is the consequence of similar flow resistances in the forward and backward directions, while the latter is caused by the surface conduction and electro-osmotic flows. ...
Flow rectification for Newtonian fluids remains challenging compared with that for non-Newtonian fluids because the physical properties of Newtonian fluids are independent of the structure of flow channels, and flow rectification can only be achieved through direction-dependent flow scenarios. In this work, we fabricate a microfluidic rectifier for Newtonian fluids using asymmetric converging–diverging microchannels. The highest diodicity measured for the rectifier is 1.77, which is 15%–54% higher than previous microfluidic rectifiers for Newtonian fluids. An expression for the diodicity is developed based on two scaling laws for the flow resistances in the forward and backward directions. Numerical simulations are also performed to confirm the experiments.
... Approaches to electroconvection modelling using the slip condition at the boundary with the SCR were applied by V. Dydek et al. [26], R. Abu-Rjal et al. [27]. Models based on the Nernst-Planck-Poisson and Navier-Stokes equations that directly take into account the formation of the extended SCR were considered in the works of E.A. Demekhin et al. [15,28,29], S.V. Pham et al. [16,30], and K. Druzgalski, E. Karatay et al. [18,31], P. Magnico [32,33]. Numerical studies of electroconvection flows generated at an electrically heterogeneous membrane surface were carried out by S. Davidson et al. [34], M. Andersen et al. [35], V.A. Kirii et al. [36]. ...
... All CVCs have a linear initial part (denoted by 1 in Figure 5a), a sloping plateau (2 in Figure 5a), and an overlimiting current (3,4 in Figure 5a), which qualitatively corresponds to the existing experimental [5,7,9,13] and theoretical [16,19,31] studies about the CVCs of membrane systems. Note that the limiting current density of the calculated CVCs, determined by the point of intersection of the tangents drawn to the initial part and to the sloping plateau of the curve is close to ilim, calculated using Leveque's Equation (29) (values differ by less than 2%) [47]: At the underlimiting and limiting current modes (regions 1 and 2 on Figure 5a, respectively) of the CVCs calculated for PD and GD regimes coincide with high accuracy (the difference is less than 0.01ilim). In these modes (at current densities iav/ilim ≤ 1 or potential drop < Vcr1), electroconvective vortices are not observed in the fluid flow (Figure 6a). ...
Electromembrane devices are usually operated in two electrical regimes: potentiodynamic (PD), when a potential drop in the system is set, and galvanodynamic (GD), when the current density is set. This article theoretically investigates the current-voltage curves (CVCs) of flow-through electrodialysis membrane systems calculated in the PD and GD regimes and compares the parameters of the electroconvective vortex layer for these regimes. The study is based on numerical modelling using a basic model of overlimiting transfer enhanced by electroconvection with a modification of the boundary conditions. The Dankwerts’ boundary condition is used for the ion concentration at the inlet boundary of the membrane channel. The Dankwerts’ condition allows one to increase the accuracy of the numerical implementation of the boundary condition at the channel inlet. On the CVCs calculated for PD and DG regimes, four main current modes can be distinguished: underlimiting, limiting, overlimiting, and chaotic overlimiting. The effect of the electric field regime is manifested in overlimiting current modes, when a significant electroconvection vortex layer develops in the channel.
... Such flow plays an important role in a wide range of applications in industrial processes, such as EHD pumps, EHD turbulent mixer, electrostatic precipitators, flow control, and heat-transfer enhancement [2][3][4][5][6][7]. Extensive studies have been devoted to this problem from the aspects of stability analysis [8][9][10], experimental studies [7,[11][12][13], and numerical simulations [14][15][16][17][18]. However, most of these studies are limited to single-phase dielectric liquids. ...
In this paper, the electroconvective flow induced by the unipolar charge injection is extended from single-phase dielectric liquid to the solid-liquid interaction problem. The physical model with fully coupled mathematical equations is built in the liquid, solid, and interface for both the Ohmic and non-Ohmic solid models. An improved lattice Boltzmann model (LBM) is developed with three lattice Boltzmann equations for Poisson's equation, charge conservation equation, and Navier-Stokes equations, respectively. Our codes are first validated by the analytical solutions at the hydrostatic state. It is found that the LBM can well reproduce the discontinuous changes of electrical field and charge density at the interface and agrees well with the analytical results. Then, simulations are conducted under different governing parameters and interface position fl. Results show that the bifurcation of electroconvection in the presence of the solid-liquid interface is still of subcritical type, but both the linear and finite amplitude stability criteria increase due to a voltage drop happening at the solid phase. Besides, the stability criterion expressed by the electrical Rayleigh number Tc increases as the permittivity ratios ɛr and the mobility ratios Kr increase, but Tc decreases with the increasing of dimensionless electric conductivity S and the interface position fl.
